where KL divergence \(D_{KL}(P_i||P_j)\stackrel{\triangle }{=}\int P_i(\mathbf {x})\ln \frac{P_i(\mathbf {x})}{P_j(\mathbf {x})}\mathrm {d}\mathbf {x}\)  and cross entropy \(H(P_i||P_j)=-\int P_i(\mathbf {x})\ln P_j(\mathbf {x})\mathrm {d}\mathbf {x}\) . The inequality (a) holds based on the variational lower bound on the expectation of a log-sum inequality \(\mathbb {E}\big [\ln \sum _i Z_i\big ]\ge \ln \big [\sum _i e^{\mathbb {E}[\ln Z_i]}\big ]\)  [1]}, and quality \((2)\)  holds based on \(H(P_i||P_j)=H(P_i)+D_{KL}(P_i||P_j)\) . As a special case for binary sensitive attribute, it is easy to obtain the following results:
\(I(\mathbf {s}, \mathbf {X})\le -(1-c)\ln \Big [(1-c) +c\exp \big (-D_{KL}(P_1||P_2)\big )\Big ]-c\ln \Big [c +(1-c)\exp \big (-D_{KL}(P_2||P_1)\big )\Big ]. \nonumber \) 
